Transmission Lines
We have obtained the following solutions for the steady-state
voltage and current phasors in a transmission line:
Loss-less line
+ − jβ z
− jβ z
V (z) = V e
Lossy line
+ −γz
− γz
+V e
(
1
I (z) =
V + e − jβ z − V − e jβ z
Z0
V (z) = V e
)
(
+V e
1
I (z) =
V + e− γ z − V −e γ z
Z0
)
Since V (z) and I (z) are the solutions of second order differential
+
−
(wave) equations, we must determine two unknowns, V and V ,
which represent the amplitudes of steady-state voltage waves,
travelling in the positive and in the negative direction, respectively.
Therefore, we need two boundary conditions to determine these
unknowns, by considering the effect of the load and of the
generator connected to the transmission line.
© Amanogawa, 2006 – Digital Maestro Series
65
Transmission Lines
Before we consider the boundary conditions, it is very convenient
to shift the reference of the space coordinate so that the zero
reference is at the location of the load instead of the generator.
Since the analysis of the transmission line normally starts from the
load itself, this will simplify considerably the problem later.
ZR
New Space Coordinate
z
d
0
We will also change the positive direction of the space coordinate,
so that it increases when moving from load to generator along the
transmission line.
© Amanogawa, 2006 – Digital Maestro Series
66
Transmission Lines
We adopt a new coordinate d = − z, with zero reference at the load
location. The new equations for voltage and current along the lossy
transmission line are
Loss-less line
+ jβd
− − jβ d
V (d) = V e
I (d) =
1
Z0
(
Lossy line
+ γd
− −γ d
+V e
+ jβd
V e
− − jβ d
−V e
V (d) = V e
)
I (d) =
1
Z0
(
+V e
+ γd
V e
− −γd
−V e
)
At the load (d = 0) we have, for both cases,
V (0) = V + + V −
(
1
I (0) =
V + −V −
Z0
© Amanogawa, 2006 – Digital Maestro Series
)
67
Transmission Lines
For a given load impedance ZR , the load boundary condition is
V (0) = Z R I (0)
Therefore, we have
(
ZR
V +V =
V + −V −
Z0
+
−
)
from which we obtain the voltage load reflection coefficient
V−
Z R − Z0
ΓR = + =
Z R + Z0
V
© Amanogawa, 2006 – Digital Maestro Series
68
Transmission Lines
We can introduce this result into the transmission line equations as
Loss-less line
Lossy line
+ jβ d
V (d) = V e
+ jβ d
I (d) =
V e
Z0
(
1+ ΓR e
(
1 − Γ Re
−2 jβ d
−2 jβ d
)
)
+ γd
V (d) = V e
+ γd
I (d) =
V e
Z0
(
1+ ΓR e
(
1 − Γ Re
−2 γ d
−2 γ d
)
)
At each line location we define a Generalized Reflection Coefficient
Γ (d) = Γ R e−2 jβ d
Γ (d) = Γ R e−2 γ d
and the line equations become
V (d) = V + e jβ d (1 + Γ (d) )
V (d) = V + e γ d (1 + Γ (d) )
V + e jβ d
I (d) =
(1 − Γ (d) )
Z0
V +e γd
I (d) =
(1 − Γ (d) )
Z0
© Amanogawa, 2006 – Digital Maestro Series
69
Transmission Lines
We define the line impedance as
1 + Γ(d)
V (d)
Z (d) =
= Z0
1 − Γ(d)
I (d)
A simple circuit diagram can illustrate the significance of line
impedance and generalized reflection coefficient:
ΓReq = Γ(d)
d
© Amanogawa, 2006 – Digital Maestro Series
ZR
Zeq=Z(d)
0
70
Transmission Lines
If you imagine to cut the line at location d, the input impedance of
the portion of line terminated by the load is the same as the line
impedance at that location “before the cut”. The behavior of the line
on the left of location d is the same if an equivalent impedance with
value Z(d) replaces the cut out portion. The reflection coefficient of
the new load is equal to Γ(d)
Γ Req = Γ(d ) =
Z Req − Z 0
Z Req + Z 0
If the total length of the line is L, the input impedance is obtained
from the formula for the line impedance as
Vin V (L)
1 + Γ(L)
=
= Z0
Z in =
1 − Γ(L)
I in
I (L)
The input impedance is the equivalent impedance representing the
entire line terminated by the load.
© Amanogawa, 2006 – Digital Maestro Series
71
Transmission Lines
An important practical case is the low-loss transmission line, where
the reactive elements still dominate but R and G cannot be
neglected as in a loss-less line. We have the following conditions:
ω L >> R
ω C >> G
so that
γ = ( jω L + R )( jω C + G )
=

R 
G 
jω L jω C  1 +
1+


j
ω
L
j
ω
C



R
G
RG
≈ jω LC 1 +
+
−
jω L jω C ω2 LC
The last term under the square root can be neglected, because it is
the product of two very small quantities.
© Amanogawa, 2006 – Digital Maestro Series
72
Transmission Lines
What remains of the square root can be expanded into a truncated
Taylor series
 1 R
G 
+
γ ≈ jω LC 1 + 

2
j
L
j
C
ω
ω



1
C
L
= R
+G
 + jω LC
2
L
C
so that
1
C
L
α = R
+G

L
C
2
© Amanogawa, 2006 – Digital Maestro Series
β = ω LC
73
Transmission Lines
The characteristic impedance of the low-loss line is a real quantity
for all practical purposes and it is approximately the same as in a
corresponding loss-less line
R + jω L
L
Z0 =
≈
G + jωC
C
and the phase velocity associated to the wave propagation is
ω
vp = ≈
β
1
LC
BUT NOTE:
In the case of the low-loss line, the equations for voltage and
current retain the same form obtained for general lossy lines.
© Amanogawa, 2006 – Digital Maestro Series
74
Transmission Lines
Again, we obtain the loss-less transmission line if we assume
R=0
G=0
This is often acceptable in relatively short transmission lines, where
the overall attenuation is small.
As shown earlier, the characteristic impedance in a loss-less line is
exactly real
L
Z0 =
C
while the propagation constant has no attenuation term
γ = ( jω L)( jω C ) = jω LC = jβ
The loss-less line does not dissipate power, because α = 0.
© Amanogawa, 2006 – Digital Maestro Series
75
Transmission Lines
For all cases, the line impedance was defined as
V (d)
1 + Γ (d)
Z (d) =
= Z0
I (d)
1 − Γ (d)
By including the appropriate generalized reflection coefficient, we
can derive alternative expressions of the line impedance:
A) Loss-less line
1 + Γ R e−2 jβd
Z R + jZ 0 tan(β d)
Z (d) = Z 0
= Z0
−2 jβd
jZ R tan(β d) + Z 0
1 − Γ Re
B) Lossy line (including low-loss)
1 + Γ R e −2 γ d
Z R + Z 0 tanh( γ d)
Z (d) = Z 0
= Z0
−2 γ d
Z R tanh( γ d) + Z 0
1 − Γ Re
© Amanogawa, 2006 – Digital Maestro Series
76
Transmission Lines
Let’s now consider power flow in a transmission line, limiting the
discussion to the time-average power, which accounts for the
active power dissipated by the resistive elements in the circuit.
The time-average power at any transmission line location is
{
1
⟨ P(d , t ) ⟩ = Re V (d) I * (d)
2
}
This quantity indicates the time-average power that flows through
the line cross-section at location d. In other words, this is the
power that, given a certain input, is able to reach location d and
then flows into the remaining portion of the line beyond this point.
It is a common mistake to think that the quantity above is the power
dissipated at location d !
© Amanogawa, 2006 – Digital Maestro Series
77
Transmission Lines
The generator, the input impedance, the input voltage and the input
current determine the power injected at the transmission line input.
Iin
ZG
VG
Vin
Generator
© Amanogawa, 2006 – Digital Maestro Series
Zin
Line
Zin
Vin = VG
ZG + Zin
1
Iin = VG
ZG + Zin
1
*
⟨ Pin ⟩ = Re Vin Iin
2
{
}
78
Transmission Lines
The time-average power reaching the load of the transmission line
is given by the general expression
{
}
1
⟨ P(d=0 , t ) ⟩ = Re V (0) I * (0)
2
 +
1
1
= Re V (1 + Γ R ) * V + (1 − Γ R )
2
Z0

(
*

) 
This represents the power dissipated by the load.
The time-average power absorbed by the line is simply the
difference between the input power and the power absorbed by the
load
⟨ P line ⟩ = ⟨ P in ⟩ − ⟨ P (d = 0 , t ) ⟩
In a loss-less transmission line no power is absorbed by the line, so
the input time-average power is the same as the time-average
power absorbed by the load. Remember that the internal impedance
of the generator dissipates part of the total power generated.
© Amanogawa, 2006 – Digital Maestro Series
79
Transmission Lines
It is instructive to develop further the general expression for the
time-average power at the load, using Z0=R0+jX0 for the
characteristic impedance, so that
R0 + jX 0 R0 + jX 0
= *
=
= 2
*
2
Z0 Z0 Z0
R0 + X 02
Z0
1
Z0
Alternatively, one may simplify the analysis by introducing the line
characteristic admittance
1
Y0 =
= G0 + jB0
Z0
It may be more convenient to deal with the complex admittance at
the numerator of the power expression, rather than the complex
characteristic impedance at the denominator.
© Amanogawa, 2006 – Digital Maestro Series
80
Transmission Lines









*
0
⟨ P(d=0, t ) ⟩ = 1 Re V + 1+Γ R  1 V + 1−Γ

Z

2
=
V+
2 Z0
=
V+
2 Z0
=
V+
2 Z0
=
V+
2 Z0
2
2





Re R0 + jX
2


0








R
+
jX
Re
0
0 
2
2












1+ Re Γ + j Im Γ





R



2
2





2






2

R0 − R0 Γ R − 2 X 0 Im Γ
© Amanogawa, 2006 – Digital Maestro Series
 
R  






=




1− Re Γ


2
2






1−   Re Γ   +  Im Γ    +






 R  
 R   










Re
+
+
2Im
R
jX
j
 1− Γ
Γ
R
0
0
2

* 
 
R  






 

R   

 





R
+ j Im Γ




  
R   



j 2Im Γ R  

 

 
=


R 

81
Transmission Lines
Equivalently, using the complex characteristic admittance:








*
0

⟨ P(d=0, t ) ⟩ = 1 Re V + 1+Γ R  Y V + 1−Γ



2
=
=
=
=
V+
2
2
2
2












1+ Re Γ + j Im Γ





R



2






 
R  






=




1− Re Γ


2
2




 




1−   Re Γ
 R   +  Im Γ R    +




 
 













2
Re G0 − jB0  1− Γ R + j 2Im Γ
2
V+







Re G0 − jB0 
2
V+


0
Re G0 − jB
2
V+





* 
 
R  

2

G0 − G0 Γ R + 2B0 Im Γ
© Amanogawa, 2006 – Digital Maestro Series





 

R   

 





R
+ j Im Γ




  
R   




 
j 2Im Γ R  

 

 
=


R 

82
Transmission Lines
The time-average power, injected into the input of the transmission
line, is maximized when the input impedance of the transmission
line and the internal generator impedance are complex conjugate of
each other.
Zin
Generator
Load
ZG
VG
ZG = Z*in
© Amanogawa, 2006 – Digital Maestro Series
Transmission line
ZR
for maximum power transfer
83
Transmission Lines
The characteristic impedance of the loss-less line is real and we
can express the power flow, anywhere on the line, as
1
⟨ P (d , t ) ⟩ = Re{ V (d) I * (d) }
2
1
 + jβ d
1 + Γ R e− j 2β d
= Re V e
2

(
)
(
1
(V + )* e− jβ d 1 − Γ R e− j 2β d
Z0
1
+2
=
V
2Z0
Incident wave
) 
*
1
+2
−
V
ΓR 2
2Z0
Reflected wave
This result is valid for any location, including the input and the load,
since the transmission line does not absorb any power.
© Amanogawa, 2006 – Digital Maestro Series
84
Transmission Lines
In the case of low-loss lines, the characteristic impedance is again
real, but the time-average power flow is position dependent
because the line absorbs power.
{
{
}
(
1
⟨ P (d , t ) ⟩ = Re V (d) I * (d)
2
1
= Re V + eα d e jβ d 1 + Γ R e−2 γ d
2
1
(V + )* eα d e− jβ d 1 − Γ R e−2 γ d
Z0
)
(
) 
*
1
1
+ 2 2α d
+ 2 −2α d
=
V
e
−
V
e
ΓR 2
2Z0
2Z0
Incident wave
© Amanogawa, 2006 – Digital Maestro Series
Reflected wave
85
Transmission Lines
Note that in a lossy line the reference for the amplitude of the
incident voltage wave is at the load and that the amplitude grows
exponentially moving towards the input. The amplitude of the
incident wave behaves in the following way
V + eα L
⇔
input
V + eα d
inside the line
⇔
V+
load
The reflected voltage wave has maximum amplitude at the load, and
it decays exponentially moving back towards the generator. The
amplitude of the reflected wave behaves in the following way
V + Γ R e−α L
⇔
input
© Amanogawa, 2006 – Digital Maestro Series
V + Γ R e−α d
⇔
inside the line
V +ΓR
load
86
Transmission Lines
For a general lossy line the power flow is again position dependent.
Since the characteristic impedance is complex, the result has an
additional term involving the imaginary part of the characteristic
admittance, B0, as
{
{
}
1
⟨ P (d , t ) ⟩ = Re V (d) I * (d)
2
1
= Re V + eα d e jβ d (1 + Γ (d) )
2
}
Y0* (V + )* eα d e− jβ d (1 − Γ (d) )*
G0 + 2 2α d G0 + 2 −2α d
=
V
e
−
V
e
ΓR 2
2
2
+ B0 V
© Amanogawa, 2006 – Digital Maestro Series
+2
e2α d Im(Γ (d))
87
Transmission Lines
For the general lossy line, keep in mind that
Z 0*
R0 − jX 0 R0 − jX 0
1
Y0 =
=
=
= 2
= G0 + jB0
*
2
2
Z0 Z0 Z0
R0 + X 0
Z0
−X0
R0
=
G0 = 2
B
0
R0 + X 02
R02 + X 02
Recall that for a low-loss transmission line the characteristic
impedance is approximately real, so that
B0 ≈ 0
and
Z 0 ≈ 1 G0 ≈ R0 .
The previous result for the low-loss line can be readily recovered
from the time-average power for the general lossy line.
© Amanogawa, 2006 – Digital Maestro Series
88
Transmission Lines
To completely specify the transmission line problem, we still have
to determine the value of V+ from the input boundary condition.
¾ The load boundary condition imposes the shape of the
interference pattern of voltage and current along the line.
¾ The input boundary condition, linked to the generator, imposes
the scaling for the interference patterns.
We have
Zin
Vin = V (L) = VG
ZG + Zin

or




with
1 + Γ (L)
Zin = Z 0
1 − Γ (L)
Z R + jZ 0 tan(β L)
Zin = Z 0
jZ R tan(β L) + Z 0
loss - less line
Z R + Z 0 tanh( γ L)
Zin = Z 0
Z R tanh( γ L) + Z 0
lossy line
© Amanogawa, 2006 – Digital Maestro Series
89
Transmission Lines
For a loss-less transmission line:
V (L) = V + e jβ L [1 + Γ (L) ] = V + e jβ L (1 + Γ R e− j 2β L )
⇒
Zin
1
V = VG
ZG + Zin e jβ L (1 + Γ R e− j 2β L )
+
For a lossy transmission line:
(
V (L) = V + e γ L [1 + Γ (L) ] = V + e γ L 1 + Γ R e−2 γ d
⇒
)
Zin
1
V = VG
ZG + Zin e γ L (1 + Γ R e−2 γ L )
+
© Amanogawa, 2006 – Digital Maestro Series
90
Transmission Lines
In order to have good control on the behavior of a high frequency
circuit, it is very important to realize transmission lines as uniform
as possible along their length, so that the impedance behavior of
the line does not vary and can be easily characterized.
A change in transmission line properties, wanted or unwanted,
entails a change in the characteristic impedance, which causes a
reflection. Example:
Z01
Z02
ZR
Γ1
Z01
© Amanogawa, 2006 – Digital Maestro Series
Zin
Zin − Z01
Γ1 =
Zin + Z01
91
Transmission Lines
Special Cases
ZR → 0 (SHORT CIRCUIT)
ZR = 0
Z0
The load boundary condition due to the short circuit is V (0) = 0
⇒ V (d = 0) = V + e jβ 0 (1 + Γ R e− j 2β 0 )
= V + (1 + Γ R ) = 0
⇒
© Amanogawa, 2006 – Digital Maestro Series
Γ R = −1
92
Transmission Lines
Since
ΓR =
V−
V+
⇒ V − = −V +
We can write the line voltage phasor as
V (d) = V + e jβ d + V − e− jβ d
= V + e jβ d − V + e− jβ d
= V + ( e jβ d − e− jβ d )
= 2 jV + sin(β d)
© Amanogawa, 2006 – Digital Maestro Series
93
Transmission Lines
For the line current phasor we have
1
(V + e jβ d − V − e− jβ d )
I (d) =
Z0
1
(V + e jβ d + V + e− jβ d )
=
Z0
V + jβ d
(e
=
+ e− jβ d )
Z0
2V +
cos(β d)
=
Z0
The line impedance is given by
V (d)
2 jV + sin(β d)
=
= jZ0 tan(β d)
Z(d) =
+
I (d) 2V cos(β d) / Z0
© Amanogawa, 2006 – Digital Maestro Series
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Transmission Lines
The time-dependent values of voltage and current are obtained as
V (d, t ) = Re[V (d) e jω t ] = Re[2 j | V + | e jθ sin(β d) e jω t ]
= 2| V + |sin(β d) ⋅ Re[ j e j (ω t +θ) ]
= 2| V + |sin(β d) ⋅ Re[ j cos(ω t + θ) − sin(ω t + θ)]
= −2| V + |sin(β d) sin(ω t + θ)
I (d, t ) = Re[ I (d) e jω t ] = Re[2 | V + | e jθ cos(β d) e jω t ] / Z0
= 2| V + |cos(β d) ⋅ Re[ e j (ω t +θ) ] / Z0
= 2| V + |cos(β d) ⋅ Re[ (cos(ω t + θ) + j sin(ω t + θ)] / Z0
| V+ |
=2
cos(β d) cos(ω t + θ)
Z0
© Amanogawa, 2006 – Digital Maestro Series
95
Transmission Lines
The time-dependent power is given by
P(d, t ) = V (d, t ) ⋅ I (d, t )
| V + |2
=−4
sin(β d) cos(β d) sin(ω t + θ) cos(ω t + θ)
Z0
| V + |2
=−
sin(2β d) sin (2ω t + 2θ)
Z0
and the corresponding time-average power is
1 T
< P(d, t ) > = ∫ P(d, t ) dt
T 0
| V + |2
1 T
=−
sin(2β d) ∫ sin (2ω t + 2θ) = 0
Z0
T 0
© Amanogawa, 2006 – Digital Maestro Series
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Transmission Lines
ZR → ∞ (OPEN CIRCUIT)
ZR → ∞
Z0
The load boundary condition due to the open circuit is I (0) = 0
V + jβ 0
⇒ I (d = 0) =
e (1 − Γ R e− j 2β 0 )
Z0
V+
=
(1 − Γ R ) = 0
Z0
⇒
© Amanogawa, 2006 – Digital Maestro Series
ΓR = 1
97
Transmission Lines
Since
ΓR =
V−
V+
⇒ V− = V+
We can write the line current phasor as
1
I (d) =
(V + e jβ d − V − e− jβ d )
Z0
1
=
(V + e jβ d − V + e− jβ d )
Z0
V + jβ d − jβ d
2 jV +
=
−e
(e
)=
sin(β d)
Z0
Z0
© Amanogawa, 2006 – Digital Maestro Series
98
Transmission Lines
For the line voltage phasor we have
V (d) = (V + e jβ d + V − e− jβ d )
= (V + e jβ d + V + e− jβ d )
= V + ( e jβ d + e− jβ d )
= 2V + cos(β d)
The line impedance is given by
V (d)
2V + cos(β d)
Z0
Z(d) =
=
=−j
I (d) 2 jV + sin(β d) / Z0
tan(β d)
© Amanogawa, 2006 – Digital Maestro Series
99
Transmission Lines
The time-dependent values of voltage and current are obtained as
V (d, t ) = Re[V (d) e jω t ] = Re[2 | V + | e jθ cos(β d) e jω t ]
= 2| V + |cos(β d) ⋅ Re[ e j(ω t +θ) ]
= 2| V + |cos(β d) ⋅ Re[ (cos(ω t + θ) + j sin(ω t + θ)]
= 2| V + |cos(β d) cos(ω t + θ)
I (d, t ) = Re[ I (d) e jω t ] = Re[2 j | V + | e jθ sin(β d) e jω t ] / Z0
= 2| V + |sin(β d) ⋅ Re[ j e j(ω t +θ) ] / Z0
= 2| V + |sin(β d) ⋅ Re[ j cos(ω t + θ) − sin(ω t + θ)] / Z0
| V+ |
= −2
sin(β d) sin(ω t + θ)
Z0
© Amanogawa, 2006 – Digital Maestro Series
100
Transmission Lines
The time-dependent power is given by
P(d, t ) = V (d, t ) ⋅ I (d, t ) =
| V + |2
=−4
cos(β d) sin(β d) cos(ω t + θ) sin(ω t + θ)
Z0
| V + |2
=−
sin(2β d) sin (2ω t + 2θ)
Z0
and the corresponding time-average power is
1 T
< P(d, t ) > = ∫ P(d, t ) dt
T 0
| V + |2
1 T
=−
sin(2β d) ∫ sin (2ω t + 2θ) = 0
Z0
T 0
© Amanogawa, 2006 – Digital Maestro Series
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Transmission Lines
ZR = Z0 (MATCHED LOAD)
Z0
ZR = Z0
The reflection coefficient for a matched load is
ZR − Z0 Z0 − Z0
ΓR =
=
=0
ZR + Z0 Z0 + Z0
no reflection!
The line voltage and line current phasors are
V (d) = V + e jβ d (1 + Γ R e−2 jβ d ) = V + e jβ d
+
V + jβ d
V
(1 − Γ R e−2 jβ d ) =
I( d ) =
e
e jβ d
Z0
Z0
© Amanogawa, 2006 – Digital Maestro Series
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Transmission Lines
The line impedance is independent of position and equal to the
characteristic impedance of the line
V (d) V + e jβ d
Z(d) =
=
= Z0
+
I (d) V jβ d
e
Z0
The time-dependent voltage and current are
V (d, t ) = Re[| V + | e jθ e jβ d e jω t ]
= | V + | ⋅ Re[e j(ω t +β d +θ) ] =| V + | cos(ω t + β d + θ)
I (d, t ) = Re[| V + | e jθ e jβ d e jω t ] / Z0
+
| V+ |
|
V
|
j (ω t +β d +θ)
=
⋅ Re[e
]=
cos(ω t + β d + θ)
Z0
Z0
© Amanogawa, 2006 – Digital Maestro Series
103
Transmission Lines
The time-dependent power is
+
|
|
V
P(d, t ) = | V + | cos(ω t + β d + θ)
cos(ω t + β d + θ)
Z0
| V + |2
cos2 (ω t + β d + θ)
=
Z0
and the time average power absorbed by the load is
1 t | V + |2
< P(d) > = ∫
cos2 (ω t + β d + θ) dt
T 0 Z0
| V + |2
=
2 Z0
© Amanogawa, 2006 – Digital Maestro Series
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Transmission Lines
ZR = jX (PURE REACTANCE)
ZR = j X
Z0
The reflection coefficient for a purely reactive load is
ZR − Z0 jX − Z0
ΓR =
=
=
ZR + Z0 jX + Z0
( jX − Z0 )( jX − Z0 )
=
=
( jX + Z0 )( jX − Z0 )
© Amanogawa, 2006 – Digital Maestro Series
X 2 − Z02
Z02 + X 2
+2j
XZ0
Z02 + X 2
105
Transmission Lines
In polar form
Γ R = Γ R exp( jθ)
where
ΓR =
(
(
)
) (
2 2
X − Z0
4 X 2 Z02
+
=
2
2
Z02 + X 2
Z02 + X 2
2
)
(
(
)
)
2
2 2
Z0 + X
=1
2
Z02 + X 2
−1  2 XZ0 
θ = tan 
 X 2 − Z2 

0
The reflection coefficient has unitary magnitude, as in the case of
short and open circuit load, with zero time average power absorbed
by the load. Both voltage and current are finite at the load, and the
time-dependent power oscillates between positive and negative
values. This means that the load periodically stores power and
then returns it to the line without dissipation.
© Amanogawa, 2006 – Digital Maestro Series
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Transmission Lines
Reactive impedances can be realized with transmission lines
terminated by a short or by an open circuit. The input impedance of
a loss-less transmission line of length L terminated by a short
circuit is purely imaginary
 2π f 
 2π 
Zin = j Z0 tan ( β L ) = j Z0 tan  L  = j Z0 tan 
L


v
 λ 
p


For a specified frequency f, any reactance value (positive or
negative!) can be obtained by changing the length of the line from 0
to λ/2. An inductance is realized for L < λ/4 (positive tangent)
while a capacitance is realized for λ/4 < L < λ/2 (negative tangent).
When L = 0 and L = λ/2 the tangent is zero, and the input
impedance corresponds to a short circuit. However, when L = λ/4
the tangent is infinite and the input impedance corresponds to an
open circuit.
© Amarcord, 2006 – Digital Maestro Series
107
Transmission Lines
Since the tangent function is periodic, the same impedance
behavior of the impedance will repeat identically for each additional
line increment of length λ/2. A similar periodic behavior is also
obtained when the length of the line is fixed and the frequency of
operation is changed.
At zero frequency (infinite wavelength), the short circuited line
behaves as a short circuit for any line length. When the frequency
is increased, the wavelength shortens and one obtains an
inductance for L < λ/4 and a capacitance for λ/4 < L < λ/2, with
an open circuit at L = λ/4 and a short circuit again at L = λ/2.
Note that the frequency behavior of lumped elements is very
different. Consider an ideal inductor with inductance L assumed to
be constant with frequency, for simplicity. At zero frequency the
inductor also behaves as a short circuit, but the reactance varies
monotonically and linearly with frequency as
X = ωL (always an inductance)
© Amarcord, 2006 – Digital Maestro Series
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Transmission Lines
Short circuited transmission line – Fixed frequency
L
Z in = 0
L= 0
0< L<
L=
λ
4
λ
2
4
λ
2
λ
k p
Im Z in > 0
in d u c ta n c e
Z in → ∞
o p e n c irc u it
k p
c a p a c ita n c e
Im Z in < 0
Z in = 0
2
< L<
L=
4
λ
< L<
L=
λ
3λ
4
3λ
4
3λ
< L< λ
4
s h o rt c irc u it
k p
s h o rt c irc u it
Im Z in > 0
in d u c ta n c e
Z in → ∞
o p e n c irc u it
k p
Im Z in < 0
c a p a c ita n c e
…
© Amarcord, 2006 – Digital Maestro Series
109
Z(L)/Zo = j tan(β L)
Transmission Lines
Impedance of a short circuited transmission line
(fixed frequency, variable length)
40
30
20
inductive
Normalized Input Impedance
10
inductive
inductive
0
-10
capacitive
cap.
-20
-30
-40
0
100
π/(2β)= λ/4
200
π/β =λ/2
300
3π/(2β)= 3λ/4
400
2π/β= λ
500
5π/(2β)= 5λ/4
θ
L
[deg]
Line Length L
© Amarcord, 2006 – Digital Maestro Series
110
Z(L)/Zo = j tan(β L)
Transmission Lines
Impedance of a short circuited transmission line
(fixed length, variable frequency)
40
30
20
inductive
Normalized Input Impedance
10
inductive
inductive
0
-10
capacitive
cap.
-20
-30
-40
0
100
vp / (4L)
200
vp / (2L)
300
3vp / (4L)
400
vp / L
500
5vp / (4L)
θ
[deg]
f
Frequency of operation
© Amarcord, 2006 – Digital Maestro Series
111
Transmission Lines
For a transmission line of length L terminated by an open circuit,
the input impedance is again purely imaginary
Z0
Zin = − j
=−j
tan ( β L )
Z0
Z0
=−j
 2π 
 2π f 
tan  L 
tan 
L
λ




v
 p 
We can also use the open circuited line to realize any reactance, but
starting from a capacitive value when the line length is very short.
Note once again that the frequency behavior of a corresponding
lumped element is different. Consider an ideal capacitor with
capacitance C assumed to be constant with frequency. At zero
frequency the capacitor behaves as an open circuit, but the
reactance varies monotonically and linearly with frequency as
1
X=
(always a capacitance)
ωC
© Amarcord, 2006 – Digital Maestro Series
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Transmission Lines
Open circuit transmission line – Fixed frequency
L
L= 0
0< L<
L=
λ
4
λ
2
4
λ
2
λ
2
< L<
L=
4
λ
< L<
L=
λ
3λ
4
3λ
4
3λ
< L< λ
4
Z in → ∞
o p e n c irc u it
Im Z in < 0
k p
c a p a c ita n c e
Z in = 0
s h o rt c irc u it
k p
Im Z in > 0
in d u c ta n c e
Z in → ∞
o p e n c irc u it
Im Z in < 0
k p
c a p a c ita n c e
Z in = 0
s h o rt c irc u it
k p
Im Z in > 0
in d u c ta n c e
…
© Amarcord, 2006 – Digital Maestro Series
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Normalized Input Impedance
Z(L)/ Zo = - j cotan(β L)
Transmission Lines
Impedance of an open circuited transmission line
(fixed frequency, variable length)
40
30
20
inductive
10
inductive
inductive
0
-10
capacitive
capacitive
capacitive
-20
-30
-40
0
0
100
π/(2β) = λ/4
200
π/β = λ/2
300
3π/(2β) = 3λ/4
400
2π/β = 2λ
500
5π/(2β) = 5λ/4
θ
[deg]
L
Line Length L
© Amarcord, 2006 – Digital Maestro Series
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Normalized Input Impedance
Z(L)/ Zo = - j cotan(β L)
Transmission Lines
Impedance of an open circuited transmission line
(fixed length, variable frequency)
40
30
20
inductive
10
inductive
inductive
0
-10
capacitive
capacitive
capacitive
-20
-30
-40
0
0
100
vp / (4L)
200
vp / (2L)
300
3vp / (4L)
400
vp / L
500
5vp / (4L)
θ
[deg]
f
Frequency of operation
© Amarcord, 2006 – Digital Maestro Series
115
Transmission Lines
It is possible to realize resonant circuits by using transmission
lines as reactive elements. For instance, consider the circuit below
realized with lines having the same characteristic impedance:
I
L1
short
circuit
Z0
V
Zin1
Zin1 = j Z0 tan ( β L 1 )
© Amarcord, 2006 – Digital Maestro Series
L2
Z0
short
circuit
Zin2
Zin2 = j Z0 tan ( β L 2 )
116
Transmission Lines
The circuit is resonant if L1 and L2 are chosen such that an
inductance and a capacitance are realized.
A resonance condition is established when the total input
impedance of the parallel circuit is infinite or, equivalently, when
the input admittance of the parallel circuit is zero
1
1
+
=0
j Z0 tan ( β r L1 ) j Z0 tan ( β r L 2 )
or
 ωr

 ωr

tan  L1  = − tan  L 2 
 vp

 vp





with
2π ωr
βr =
=
λ r vp
Since the tangent is a periodic function, there is a multiplicity of
possible resonant angular frequencies ωr that satisfy the condition
above. The values can be found by using a numerical procedure to
solve the trascendental equation above.
© Amarcord, 2006 – Digital Maestro Series
117